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Newton's method is a method for approximating the value of the roots of a function that cannot be solved for algebraically. Given the function f(x) and an estimate value for the root x0, the first approximation is

$x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}$

The second is

$x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}$

and in general

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

The more times this process is repeated, the better the approximation will be.

For best results, Newton's method should be initiated where $f'(x_0)$ is sufficiently large, $f''(x_0)$ is sufficiently small, and $x_0$ is close to the intended root. This will help avoid certain situations involving unexpected roots, or non-converging situation.

## Example

Suppose we are given the function

$f(x) = e^{3x} + x^3$

We will start with the approximation x0 = -0.5. The first approximation will be

$x_1 = -0.5 - \frac{f(-0.5)}{f'(-0.5)} = -0.5 - \frac{e^{-1.5} + (-0.5)^3}{3e^{-1.5} + 3(-0.5)^2}= -0.569$

The second will be

$x_2 = -0.569 - \frac{f(-0.569)}{f'(-0.569)} = -0.569 - \frac{e^{-1.5} + (-0.569)^3}{3e^{-1.5} + 3(-0.569)^2}= -0.567$

Plugging this into the original equation, we get

$f(-0.567) = e^{3 \cdot -0.567} + (-0.567)^3 = 0.000217$

The more approximations we make, the closer to zero the function will become.

## Damped Newton's Method

To help improve convergence, newton's method may be dampened with a constant α from (0,1].

$x_{n+1} = x_n - \alpha_n\frac{f(x_n)}{f'(x_n)}$

Ideally, the each value of α should have the next iteration get as close to the root as possible. Ony possible method of determing α is the Band-Rose algorithm.

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